The physical significance of this apparent curvature of the world-line of the body will be that the body will now appear to possess acceleration with respect to our new non-Galilean frame. The precise magnitude of this acceleration will therefore be given by the apparent curvature of the world-line followed by the body, and this curvature is of course expressed implicitly by the equation of the world-line when referred to our curvilinear mesh-system. But for a given test-body the acceleration is related to the force acting on the body. From this it follows that the mathematical expression of the force can be obtained immediately.
This mathematical expression of the curved world-line, hence of the force of inertia at any particular point, is seen to be built up with the variations in value of the
’s around this point. Were the
’s to remain constant in value throughout, as would be the case in a Galilean frame, this mathematical expression would vanish in value. It follows that the
’s, of which this expression of the force is built, must correspond to the potentials of inertia. We have thus discovered the physical significance of the
’s of space-time: they define potentials.